Classical Mathematical Models for Description
and Forecast of Preclinical Tumor Growth
Sébastien Benzekrya,b, Clare Lamonta, Afshin Beheshtia, Lynn Hlatkya,
Philip Hahnfeldta
aCenter
of Cancer Systems Biology, GRI, Tufts University School of Medicine,
Boston, MA, USA
bInria
Bordeaux Sud-Ouest, Institut de Mathématiques de Bordeaux, Bordeaux,
France
Corresponding author: S. Benzekry, [email protected]
Word count:
Abstract: 266 (300)
Author summary: 193 (200)
Body text: 6240
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Abstract
Tumor growth is a complex process involving a large number of biological
phenomena. However, at the macroscopic scale, it seems to follow relatively simple
laws that have been formalized with the help of mathematical models.
Based on experimental data of in vivo syngeneic tumor growth, rigorous quantitative
and discriminant analysis of a wide array of these models was performed for
description and prediction of tumor kinetics. Detailed analysis of the measurement
error was performed that resulted in the design of an adapted statistical model of the
variance error quantifying the uncertainty of the data. Combined to several goodnessof-fit criteria and to a study of the numerical identifiability of the models, it allowed
quantification of the descriptive power of each model, from which we inferred insights
on macroscopic tumor growth laws. Our analysis enlightens one model as particularly
relevant, namely the power growth model, which suggests a novel, simple and
minimal theory of neoplastic development based on a fractal dimension of the
proliferative tissue.
Detailed study of the predictive properties of the models reveals variable forecasting
power among them and quantifies how far and how precise predictions can be made,
based on a given number of data points. In situations where small number of data
points is available, we studied the effect of adjunction of a priori information on the
statistical distribution of the parameters during the fit procedure. This method
revealed very powerful, yielding significant improvement of the forecast
performances, for instance from a 14.9% to a 60.2% success rate when predicting
future growth based only on three data points and using the power growth model.
Author Summary
Although depending on a wide array of intricate phenomena, tumor growth results, at
the macroscopic scale, in relatively simple time curves that can be quantified using
mathematical models. Here we assessed the descriptive and predictive power of the
most classical of these in order to infer general laws for the global behavior of
neoplastic growth. As a result from our analysis one of the models, namely the power
growth model, appears as particularly adapted and proposes a novel theory of tumor
growth based on a fractal dimension of the proliferative tissue.
We also assessed the predictive power of these classical mathematical models and
show that despite similar descriptive accuracy, models can significantly differ in their
ability to predict future growth. When only few data points are available, we propose
to use some a priori information during the estimation process, a method that reveals
very helpful and significantly improves the prediction success rate.
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These results could be of value for preclinical cancer research by suggesting what
model is best adapted when assessing anti-cancer drugs efficacies. They also offer
clinical perspective on what can be expected from mathematical modeling in terms of
future growth prediction.
Introduction
Neoplastic growth involves a large number of complex biological processes, including
control of the cell cycle, stroma recruitment, angiogenesis or escape from immune
surveillance, that in combination produce a macroscopic expansion, thus raising the
prospect of a possible general law for the global dynamics of neoplasm. Quantitative
and qualitative aspects of the temporal development of tumor growth can be studied
in a variety of experimental settings, including in vitro proliferation assays, threedimensional in vitro spheroids, syngeneic or xenograft in vivo implants (subcutaneous
or at the orthotopic site), transgenic mice models and longitudinal studies of clinical
images. Each scale has its own advantages and drawbacks, with increasing
relevance tending to coincide with decreasing measurement precision. Here we focus
on a model of syngeneic Lewis Lung carcinoma (LLC) subcutaneously implanted in
immune-competent mice.
Global quantification of tumor cell kinetics is a full discipline, termed cytokinetics [1],
that can be addressed using mathematical models whose goals are twofold: 1) the
assessment of global growth theories through hypotheses testing and 2) the forecast
of future/past growth based on restricted number of data [2]. While the former is
useful to uncover possible general mechanisms governing macroscopic tumor
growth, the latter can be applied in preclinical drug development [3–5] or even for
clinical personalized prediction of further disease development (examples of such
applications can be found in [6,7] for prognosis of lung metastases, in [4,8,9] for
studies on gliomas or in [10] for prostate cancer response to androgen suppression
therapy). Cancer modeling offers a wide range of mathematical models (see [11] for
a historical perspective) that can be classified according to their scale, approach
(bottom-up versus top-down) or integration of spatial structure. At the cellular scale,
agent-based models (see for instance [12]) are well-suited for studies on tumor
spheroids, while the tissue scale is better described by continuous partial differential
equations like reaction-diffusion models [9,13] or continuum-mechanics based
models [14,15]. Here we focus on scalar data of volume longitudinal development
and will consider non-spatial models for macroscopic description of tumor kinetics,
based on ordinary differential equations. A plethora of models exists at this scale
(see [16]), which reduces tumor growth to its more essential components, the most
essential being pure proliferation (leading to exponential growth). Observations of
non-constant doubling time during tumor history [17] have led investigators to
consider more elaborated models such as the widely used Gompertz model [18,19],
logistic or generalized logistic models, power growth [20] or two phase models
(exponential followed by linear, see [5]). More recently, new models integrated tumor
3
neo-angiogenesis in the modeling by considering a dynamical carrying capacity
[3,21].
Existence of this broad class of models raises the question of their relative relevance
for description and forecast of tumor growth within a given preclinical mouse model.
Here we present a comparative study of these models for both their descriptive and
predictive power.
Materials and Methods
Ethics statement
This study was performed in strict accordance with the recommendations in the
Guide for the Care and Use of Laboratory Animals of the National Institutes of Health.
The protocol was approved by the Institutional Animal Care and Use Committee
(Protocol: #P11-324). The institution is AAALAC accredited and every effort was
made to minimize suffering to the mice involved.
Mice experiments
Cell culture
Murine Lewis lung carcinoma (LLC) cells, originally derived from a spontaneous
tumor in a C57BL/6 mouse [22], were obtained from American Type Culture
Collection (Manassas, VA). The LLC cells were cultured under standard conditions
[22] in high glucose DMEM (Gibco Invitrogen Cell Culture, Carlsbad, CA) with 10%
FBS (Gibco Invitrogen Cell Culture) and 5% CO2.
Tumor Injections
C57BL/6 male mice were used with an average lifespan of 878 days [23]. At time of
injection mice were 6 – 8 weeks old (Jackson Laboratory, Bar Harbor, Maine).
Subcutaneous injections of 106 LLC cells in 0.2 ml phosphate-buffered saline (PBS)
were performed on the caudal half of the back in anesthetized mice. Tumor size was
measured regularly with calipers to a maximum of 1.5 cm 3 when mice were sacrificed
and the tissues processed.
Tissue processing
Mice were sacrificed with a 0.6 ml intraperitoneal injection of 2,2,2-tribromoethanol at
20 mg/ml. Tissues to be frozen-sectioned were dissected and slow-frozen in OCT
(Tissue Tek, Fisher Scientific, Pittsburgh, PA) in the gas phase of liquid nitrogen.
Tissues to be paraffin-sectioned were placed in 10% formalin, processed by standard
protocol (R. D. Lillie, Histopathologic technic and practical histochemistry. Blakiston
Division, New York, 1965.), placed in cassettes, and paraffin-embedded. Paraffinembedded tissues were cut into 4 μm slices, placed on positively charged slides
(Fisher Scientific), and stained for hematoxylin and eosin (H&E) stain using standard
4
protocols (R. D. Lillie, Histopathologic technic and practical histochemistry. Blakiston
Division, New York, 1965.).
Models
The simplest model used for description of tumor growth is exponential growth,
representing proliferation of a constant fraction of the total number of cells with
constant cell cycle length. Initial exponential phase can be assumed to be followed
by a linear growth phase, as done in [5]. The associated differential equation for the
volume rate of change (growth rate) is
𝑑𝑉
= 𝑎0 𝑉,
𝑡≤𝜏
𝑑𝑡
𝑑𝑉
= 𝑎1 ,
𝑡>𝜏
𝑑𝑡
{ 𝑉(𝑡 = 0) = 𝑉0
where coefficient 𝑎0 is related to proliferation (it is the fraction of proliferative cells
times the inverse of the cell cycle length), 𝑎1 drives the linear phase, time 𝜏 is
1
𝑎1
computed such that there is continuity of the derivative (𝜏 = 𝑎 log (𝑎
0
0 𝑉0
)) and 𝑉0 is
the initial volume. We considered three models deriving from this formula: a) fixed
initial volume 𝑉0 = 106 𝑐𝑒𝑙𝑙𝑠 = 1 𝑚𝑚3 (number of injected cells) and no linear phase
(𝑎1 = +∞), referred to hereafter as exponential 1, b) free initial volume and no linear
phase, referred to as exponential 𝑉0 and c) the full model, referred to as the
exponential-linear model.
Other growth models were considered, with the intent to fit the data without letting the
initial volume as a degree of freedom and fixing its value to the number of injected
cells). Although the number of cells that actually take to form a tumor is probably
lower than the number of injected cells (around 60-80%), we consider 1 𝑚𝑚3 as a
reasonable approximation considering that the order of magnitude of the total growth
curve is 1000 𝑚𝑚3 . The Gompertz model is described by the following differential
equation and initial condition
𝑑𝑉
𝐾
= 𝑎𝑉 log ( )
{ 𝑑𝑡
𝑉
𝑉(𝑡 = 0) = 1 𝑚𝑚3
where 𝑎 is a coefficient related to proliferation kinetics and 𝐾 is the so-called carrying
capacity (maximal reachable volume). This model is built-up to exhibit exponential
decrease of the relative growth rate (defined as
5
1 𝑑𝑉
𝑉 𝑑𝑡
). The Gompertz model
corresponds to the limit when 𝛼 goes to zero of the generalized logistic model
𝑎
(providing parameter 𝑎 is rescaled to 𝛼):
𝑑𝑉
𝑉 𝛼
= 𝑎𝑉 (1 − ( ) )
{ 𝑑𝑡
𝐾
𝑉(𝑡 = 0) = 1 𝑚𝑚3
When 𝛼 = 1, this model will be simply referred to as the logistic model. In this case
𝑉
the instantaneous probability of proliferation of a cell is proportional to 1 − 𝐾, which
corresponds to mutual competition between the cells, due to space or nutrients
limitation, for instance.
One complexity step above is a model assuming a dynamic (time-dependent)
carrying capacity and introduced in [3,21]. The underlying theory consists in
interactions of the tumor with its vasculature, represented by the carrying capacity
(CC) 𝐾. Additionally considering that stimulation of the carrying capacity is
proportional to the tumor surface, this model writes
𝑑𝑉
𝐾
= 𝑎𝑉 log ( )
𝑑𝑡
𝑉
𝑑𝐾
= 𝑏𝑉 2/3
𝑑𝑡
{𝑉(𝑡 = 0) = 1 𝑚𝑚3 , 𝐾(𝑡 = 0) = 𝐾0
and will be referred to as the dynamic CC model.
Eventually we considered the following model, first introduced for tumor growth
description in [20] and that will be referred to as the power growth model
𝑑𝑉
= 𝑎𝑉 𝛾
{
𝑑𝑡
𝑉(𝑡 = 0) = 1 𝑚𝑚3
The underlying theory is that only a subset of the cancer cells are cycling, that part
being proportional to a power of the volume, and thus having a smaller, possibly
fractional, dimension than the tumor itself, proposing the concept of
2
subdimensionality of the proliferative tissue. For instance 𝛾 = 3 could represent a
proliferative rim limited to the surface of the tumor, while 𝛾 = 1 describes proliferative
cells uniformly distributed within the neoplasm and recovers exponential growth. It
should be noted that in the former case simple calculations show that the tumor
radius (proportional to 𝑉 2/3 ) grows linearly in time. A more complex equation than the
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power growth model that consists in adding a loss term proportional to the volume in
the differential equation, known as the von Bertalanffy model [24], will not be
considered here because it resulted, in our analysis, in higher Akaike Information
Criterion than the mere power growth equation (1073 versus 1070).
Fit procedure and goodness of fit criteria
Weighted least-squares
Based on statistical analysis of the measurement error (see the Results section), we
fitted the models to the data using weighted nonlinear least square minimization. The
following objective was used
𝑛
(1)
(𝑌𝑖 − 𝑓(𝑡𝑖 , 𝛽))
𝐽(𝛽) = ∑
𝛴𝑖2
2
𝑖=1
where 𝑓(𝑡𝑖 , 𝛽) stands for the output of model 𝑓 at time 𝑡𝑖 with parameter set 𝛽, 𝑛 is
the number of measurements and 𝛴𝑖 is the variance associated to measure error for
𝜎𝑌 𝛼 ,
𝑌𝑖 ≥ 𝑉𝑚
𝑌𝑖 given by 𝛴𝑖 = { 𝑖 𝛼
(see Results). Minimization of the sum of
𝜎𝑉𝑚 ,
𝑌𝑖 < 𝑉𝑚
squared residuals was performed using the built-in Matlab [25] function lsqcurvefit
that is based on a trust-region reflective optimization algorithm. Bounds on
parameters were imposed, but only to ensure positivity of the parameters. Values of
the parameters used for initialization of the minimization algorithm are reported in the
supplementary Table 1.
Normalized error (𝑁𝐸)
From the obtained fit, various indicators of the goodness of the fit can be defined.
Based on our a priori analysis of the measurement error we define the normalized
error as
𝑌𝑖 − 𝑓(𝑡𝑖 , 𝛽)
𝑁𝐸𝑖 = |
|
𝛴𝑖
for measurement 𝑖. We then considered the median of these normalized errors over
all the fitted data points, pooling all the animals together. The resulting criterion
(median normalized, denoted 𝑚𝑁𝐸) quantifies, in number of standard deviations, the
median relative position of the model simulation within the errorbars and is
considered as a good indicator of the goodness of fit. Notice that this criterion is
relative to our underlying error model.
Coefficient of determination
The coefficient of determination is defined by
2
𝑅 =1−
∑𝑛𝑖=1(𝑌𝑖 − 𝑓(𝑡𝑖 ))
∑𝑛𝑖=1(𝑌𝑖 − 𝑌)
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2
2
where 𝑌 is the time average of the data points. It quantifies how much of the
variability in the data is described by the model and how better is the model than
fitting the data by only a constant line equal to the average value.
Root mean squared error (RMSE)
Another classical goodness of fit criterion that also penalizes lack of parsimony of the
model (i.e. too many parameters) is given by the Root Mean Squared Error (RMSE)
defined by
𝑅𝑀𝑆𝐸 = √
1
𝐽(𝛽 ∗ )
𝑛−𝑝
where 𝑝 is the number of parameters and 𝛽 ∗ is the parameter set giving the best fit.
p-value
We performed statistical goodness of fit 𝜒 2 test (performed using 20 bins) to evaluate
if the weighted residuals from a given fit with a given model were normally distributed
with a standard deviation being the one observed in the measurement error analysis
(𝜎 = 0.21).
Population approach, mixed-effect statistical models and a priori
information
Mixed-effect statistical models
The approach we explained above for the fitting procedure is based on minimization
of criterion defined by formula (1) for one given individual and does not account for
the fact that all the individuals are part of a same population and thus should
somehow react the same, although having a (possibly wide) inter-animal variability.
The mixed-effect approach, implemented in the Monolix software [26], consists in
pooling all the individuals together and estimate a global distribution of the model
parameters in the population. It is based on maximizing the likelihood of a
parameterized distribution, which is done using the SAEM algorithm (stochastic
algorithm for global optimization). From the estimation of the global log-likelihood
(LLH), standard goodness of fit scores are derived such as the Akaike Information
Criterion (AIC), defined as
𝐴𝐼𝐶 = −2 ∗ 𝐿𝐿𝐻 + 2 ∗ 𝑝
where 𝑝 is the number of parameters in the model. Such criterion allows ranking of
the models for their fitting power, taking into account parsimony. Overall, the results
8
we obtained using Monolix where similar to the ones we had using individual fits (see
Table 2).
A priori information
For tumor growth forecast, we used a statistical population approach considering
integration of a priori information on the population distribution of parameters. We
integrated 10 additional mice from a different but similar study of subcutaneous LLC
growth in C57/BL6 mice, pooled all the animals together and randomly divided them
into two groups. For each model, on the first group, individual fits where performed
using all the available data. This allowed us to derive mean and standard deviation of
the models parameters within the population. We then used this information to
penalize the sum of squared residuals used for fits of the second group individuals, in
the following way
𝑛
2
𝑝
(𝛽𝑗 − 𝛽̂𝑗 )
(𝑌𝑖 − 𝑓(𝑡𝑖 , 𝛽))
1
1
𝐽𝑃 (𝛽) = ∑
+
∑
𝑛
𝑝
𝛴𝑖2
𝜔𝑗2
𝑖=1
2
𝑗=1
where 𝛽̂𝑗 is the mean value of parameter 𝛽𝑗 ’s population distribution, 𝜔𝑗 is its
standard deviation and 𝑝 is the number of parameters. This procedure was repeated
100 times (i.e. 100 random assignments of the total population between 10 “learning”
animals and 10 “forecast” animals), this number being considered as sufficient to be
in the convergence limit of the large numbers law (no significant difference between
20 and 100 replicates, 𝑝 > 0.2 by Student’s t-test).
Results
Measurement error
We estimated the measurement error due to imprecision when using calipers in
assessment of the tumor volume. Some of the measurements, one per time point per
cage (over a total of 12 time points and 8 cages), were done twice within a few
minutes interval in order to estimate the variability in the measurement. This gave in
total 133 measures for which we have information about the error, which can be
analyzed by considering the following statistical representation
𝑌 = 𝑌𝑇 + 𝛴𝜀
where 𝑌 stands for the measure of a tumor whose real volume is denoted by 𝑌𝑇 , 𝜀 is
a reduced centered Gaussian random variable and 𝛴 is the error variance. The two
independent measures we performed, termed 𝑌1 and 𝑌2 , are strongly correlated
(Figure 1A, correlation coefficient 𝑟 = 0.98), but statistical analysis rejects variance
independent of volume (𝑝 = 0.004, 𝜒 2 test) and simple proportional error is only
weakly significant (𝑝 = 0.083, Figure 1B). Indeed, errors made on small tumors are
underestimated when considered proportional to the volume due to the difficulty to
9
detect the edges of subcutaneous implants. On the other hand large measurement
errors are overestimated with a proportional variance. To overcome these two issues
we propose the following expression of the error variance
(2)
𝜎𝑌 𝛼 ,
𝛴={ 𝛼
𝜎𝑉𝑚 ,
𝑌 ≥ 𝑉𝑚
𝑌 < 𝑉𝑚
In this model variance of the error is proportional to a power 𝑌 𝛼 of the volume for
tumor volumes larger than 𝑉𝑚 while volumes smaller than 𝑉𝑚 have same error as
measuring 𝑉𝑚 . The proportionality coefficient is denoted by 𝜎,. We explored various
values of 𝑉𝑚 and 𝛼 and found 𝛼 = 0.84, 𝑉𝑚 = 83 𝑚𝑚3 to be able to describe
dispersion of the error (𝑝 = 0.196, Figure 1C). This yielded a value of 𝜎 = 0.21.
This result was further confirmed by a fitting analysis performed with Monolix
software [26], in which the Akaike information criterion (AIC) criterion was found to be
lower when using an error variance model proportional to a power of the volume, as
compared to constant or proportional error models (see Methods for details about
mixed-effect models and population approach that is implemented in Monolix).
These results allow precise quantification of the measurement error inherent to our
data, which is a fundamental step towards the assessment of a model’s descriptive
power since it allows us to quantitatively determine whether the data we have could
have been generated by the model.
Robustness and numerical identifiability of the models
Specific assumption
For the dynamic CC model, allowing the stimulation power to vary did not significantly
improve fit performances and resulted in higher Akaike Information Criterion,
2
justifying the assumption of a fixed power (𝛾 = 3).
Numerical identifiability
Numerical identifiability of the fits to the initial parameter set given to the trust-region
reflective minimization algorithm used by Matlab was assessed by systematically
varying initialization of the algorithm. Compact subset of the parameter space of
length 4 standard deviations above and below a baseline mean value (obtained by
an a priori fit) in each parameter direction was meshed (11 discretization steps for
each direction) and explored, with a total of 10 × 11𝑃 individual fits performed for
each model, with 𝑝 being the number of parameters in the model. We report in Table
1 results of sensitivity scores, defined as the fraction of fits that converged to the
same parameter set as the baseline value (within a range of 10% error). When global
numerical identifiability was not observed, further study was performed and the
resulting variation of the best-fit parameter sets computed (Table 1).
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The dynamic CC model exhibited a low identifiability score of 61.5% (out of 13310
fits). This is mostly due to its bi-dimensional nature (volume and carrying capacity are
variables) with only one observable used for the fits. This fact results in variability
mostly in estimation of 𝐾0 with large deviations although significant variations were
obtained also in the other parameters. The generalized logistic model had very low
identifiability score (1.19%), mostly due to high volatility of parameter 𝛼 (median
deviation 390%) that confirms tendency of this parameter to be close to zero in our
data and suggests the Gompertz as more adapted in the class of sigmoid-like
models.
All the other models exhibited very good robustness in the parameter estimation,
specifically the power growth model for which 100% of the fits gave the same optimal
parameter set as the one derived with the baseline initial guesses, suggesting global
numerical identifiability of these models.
Goodness of fit
We tested all the models for descriptive properties and quantified the goodness of fit
on the total population of growth kinetics according to various criteria (see
supplementary material for their definition). Results are reported in Table 2 and
Figure 2.
The exponential 1 model was not able to describe the kinetics of our data (Figure 2B,
Table 2) while allowing the initial volume to be a free parameter resulted in much
better descriptive properties (Figure 2B). However, this resulted in a significantly
higher initial volume 𝑉0 than the number of injected cells (23.4 ± 7.47 𝑚𝑚3 in the fits
versus 1 𝑚𝑚3 injected) and thus provided a biologically unrealistic description of
tumor growth. Hence, in the following we did not allow for a free 𝑉0 arguing that: a) it
results in implausible initial volumes and b) models are usually very sensitive to this
initial condition as small difference propagates exponentially in time. Consequently,
for all the other models we fixed 𝑉0 = 1 𝑚𝑚3 .
Although the exponential-linear model exhibited good fits to the data in [5] it was not
able to accurately describe ours (Figure 2B) as we don’t observe stabilization of the
growth in a linear regimen. Similarly, the logistic model had poor descriptive power
(Figure 2B and Table 2). The three models exponential 𝑉0, exponential-linear and
logistic can be grouped together regarding to their fit properties, as models with
approximate descriptive power. Indeed, these three models all failed the statistical 𝜒 2
goodness of fit test (𝑝 < 0.001, Table 2). It should also be noted that the R2 score
appears as a coarse value of the fitting power because, although these models have
R2 values larger than 0.95, fits remain unsatisfactory as expressed by a median value
of the normalized error larger than 1.
All the goodness of fit criteria we considered globally performed similarly and allowed
us to rank the models (in ascending order for AIC in Table 2) for their descriptive
power. It is worth noting that goodness of fit results were globally consistent, whether
11
using the Matlab built-in algorithm for minimization of least squares (based on a
deterministic descent method) or the stochastic approximation of expectation
maximization (SAEM) algorithm for likelihood maximization implemented in Monolix
and based on a mixed-effect statistical population approach.
Four models (dynamic CC, Gompertz, power growth and generalized logistic) did not
fail the test of Gaussian distribution of residuals with standard deviation given by the
measurement error analysis (p>0.05, Table 2) and fell within the error bar for more
than half of all the data points (𝑚𝑁𝐸 < 1, Table 2). Apart from the generalized logistic
growth that had relatively elevated mean root mean square error (higher than
exponential 𝑉0), the other three models exhibited excellent scores for all the criteria
considered; the slight differences between them did not allow discrimination between
them. Parameter 𝛼 from the generalized logistic was globally estimated to a very low
value (0.06 ± 0.13, Table 3), suggesting a trend toward the Gompertz model since
the generalized logistic converges to the Gompertz when 𝛼 goes to 0. Indeed, the
Gompertz has better AIC and mean RMSE as it performed better with fewer degrees
of freedom.
Interestingly, two of the “high descriptive power” models have only 2 parameters
(power growth and Gompertz); suggesting that the data we dispose intrinsically has
less than two degrees of freedom.
Despite the complexity of internal cell populations and tissue organization (Figure
2A), these results show that at the macroscopic scale tumor growth exhibits relatively
simple dynamic that can be captured through mathematical models. However, not all
of them are equal in terms of descriptive power. Exponential based and logistic
models have to be rejected while the power growth, Gompertz, dynamic CC and
generalized logistic could reasonably have generated our data. These last four
models are indistinguishable from one another and represent valid mathematical
theories for the description of in vivo tumor growth.
Insights on macroscopic growth laws from mathematical models
From the observation that fitting to the exponential function requires assuming an
unrealistically large initial volume parameter 𝑉0 , we deduce that growth occurs at a
faster rate at initiation than at later time points. This is substantiated by the
observation that computed growth rates from the exponential 𝑉0 model are
significantly lower than reported for the in vitro growth rate of LLC cells (𝑎 = 0.233 ±
0.0164 𝑑𝑎𝑦 −1 in our analysis versus 1.12 ± 0.12 𝑑𝑎𝑦 −1 in vitro in [27]). Furthermore,
although exponential growth gives a reasonable first-order approximation of
established growth, superior fitting power of models exhibiting a decrease of the
relative growth rate such as the Gompertz, generalized logistic or dynamic CC (Table
2) suggests non-constant doubling time over growth history. Although the logistic
model does incorporate a non-constant doubling time feature, it did not accord with
the data, while the Gompertz model did. This suggests that the underlying hypothesis
12
of the logistic model - a slowdown of the relative growth rate due only to competition
between the cancer cells – is not sufficient to explain the observed growth. On the
other hand, gompertzian growth (i.e., exponential decay of the relative growth rate)
appears suitable as a descriptive model for this slowdown, although not providing any
biological insight into the mechanistic basis for this observation. All the four best
descriptive models, power growth, Gompertz, dynamic CC and generalized logistic,
exhibited very similar relative growth rate decay profiles (supplementary Figure 1),
emphasizing a possible general law in the way the fraction of proliferative cells
decreases within tumors over time.
Due in parts to excellent identifiability of the model, the power growth parameters
resulting from the fit exhibit low inter-animal variability, in particular coefficient 𝛾.
The power growth model offers a theory that integrates linear growth of the tumor
diameter, reported for instance in the case of gliomas [28]. This situation naturally
2
occurs when 𝛾 = 3 and describes proliferative cells limited to the surface of the lesion.
On the other hand, exponential growth is recovered when 𝛾 = 1. However the model
does not limit to these two extreme cases as any fractional power can occur. In our
results, the fractal dimension of proliferative tissue was found to lie between 2 and 3
(𝛾 = 0.74 ± 0.05, giving a fractal dimension of 2.23 ± 0.15). This observation is
substantiated by experimental results [3] where proliferative cells were found within
the hypoxic regions of experimental tumors lying in the interior of the tumors. Our
value of 𝛾 also matches remarkably well the values reported in [20] where the power
growth model was shown to accurately fit to growth data of 300 C3H mouse
mammary tumors (95% confidence interval of 0.73 ± 0.08 in their analysis versus
0.75 ± 0.03 in ours), suggesting a similar fractal pattern for the two different cell
lines. Moreover, the relative growth rate at injection (given by the other model
parameter, 𝑎) that we inferred from the fits is in remarkable agreement with its in vitro
counterpart, the proliferation rate of LLC cells (1.11 ± 0.23 𝑑𝑎𝑦 −1 in our analysis
versus 1.12 ± 0.12 𝑑𝑎𝑦 −1 in vitro [27], a value confirmed by proliferation kinetics
observed in our laboratory). Hence the power growth model offers not only good
description of the growth curves of subcutaneous tumor growth but also a biologically
meaningful law for neoplastic development with relevant coefficient values. Based on
a parsimonious expression, it reproduced the observed and already reported decay
of the relative growth rate and elucidates the phenomenon in terms of a plausible
biological explanation.
The power growth model thus offers a valid and simple theory of macroscopic
neoplastic growth that highlights a possible fractal nature of the proliferative tissue.
Forecasting tumor growth
The three highly descriptive models power growth, Gompertz and dynamic CC, were
further assessed for their predictive power. Adjunction of the exponential 𝑉0 model
13
was also considered. The challenge considered was to estimate future growth based
on a given number of data points.
Models’ predictive power
Despite similar descriptive properties (Table 2), the models did not perform equally
well with regard to prediction (Figure 3). The first setting we considered was to
predict future growth based parameters fitted using six data points. Goodness of the
prediction was quantified by the median normalized error between model predictions
and data, over all the future data points (Table 3). We also tested the predictability of
the next data point (Table 3). Figure 3 illustrates the predictive performances of all
the models for a given mouse. Model prediction was considered successful when
median NE over the remaining data points was lower than three, corresponding to a
model prediction within three standard deviations of the measurement error of the
data. This methodology was considered in agreement with direct visual assessment
of the goodness of prediction (see supplementary Figure 2 where all the individual
dynamic CC predictions are presented). In this setting, the dynamic CC model gave
good results, being able to predict 70% of the animals’ future trends and 90% of them
at horizon (depth) 1 (Table 4). Situations where the model failed to accurately predict
occurred when unexpected growth acceleration (animal 4 in supplementary Figure 2)
or deceleration (animal 6 in supplementary Figure 2) occurred. On the other hand,
the good descriptive properties of the Gompertz model did not translate into accurate
predictive power since less than 50% of the individuals could be globally predicted
(Table 4). The power growth model exhibited satisfactory predictive power (60%
success rate for global future growth and 70% at horizon 1, see Table 4) despite its
low number of parameters.
Prediction depth
For evaluation of the global predictive properties of the models, predictions using a
variable number of given data points and varying prediction depth were performed
and are shown in Figure 4 and Table 4. The number of data points used in a
particular setting is denoted by 𝑁, with 𝑁 ranging from 3 to 7. We then asked the
models to predict the 𝑑-th next data point (i.e. the 𝑁 + 𝑑-th one), with the prediction
depth 𝑑 ranging from 1 to 7. The associated success rate for a given model was
denoted 𝑆𝑑𝑁 . The results are summarized in Figure 4. Only two models had an overall
median success rate higher or equal to 50%, namely dynamic CC (55%) and power
growth (60%). The better median overall success rate of the power growth model is
compensated by a lower mean overall success rate (48.3% versus 51.3%) indicating
that, in those situations where the dynamic CC model was able to predict, it had a
higher success rate. This happened for instance when 𝑁 = 4 or 𝑁 = 6. Globally, for
the power growth and dynamic CC models, the high descriptive accuracy translated
into substantial predictive power. Not surprisingly, the success rate increases when
the models are fed with more data points or when the prediction depth is reduced.
14
In contrast, despite its very good descriptive properties (Table 2 and Figure 2), the
Gompertz model exhibited low global predictive power, with a median overall success
of 27.5% (Table 4 and Figure 4) and a lower score than the power growth in all
situations but one (𝑁 = 6 and 𝑑 = 1).
Interestingly, the basic exponential model with free initial volume performed well at a
short time depth, even reaching a 90% success rate for prediction of the next data
point using only 3 data points, while power growth and dynamic CC had the same
score of only 60% in this situation. This can be explained by the mathematical fact
that locally every dynamical system is exponential. Moreover the exponential model
cannot exhibit slowdown of the relative growth rate. Thus, while models such as the
power growth and the dynamic CC interpret an initial slow growth (or even stable
initial growth) as a possible very fast decrease of the growth rate, the exponential
does not and is able to predict an unexpected larger fourth time point. However,
when looking at depth 2, the exponential model looses its superiority with the
success rate 𝑆23 drastically falling to 50%, while the power growth is stable at 60%
and dynamic CC falls to 50%.
Taken together, these results suggest the power growth and the dynamic CC models
as appropriate candidates for tumor growth forecast. Within the setting considered in
this section, predictions can be made with an accuracy level of approximately 60%,
up to a depth of four days (corresponding to approximately five months in humans)
with sufficient number (≥ 6) data points, while only a two days depth can be
expected when fewer data points are used.
A priori information
When few number of data points are available, for example with only three, individual
predictions based on fits only was shown to be globally poor, especially over a large
time frame (Figure 4). However, this situation is likely to be the clinically relevant
since few clinical examinations are performed before the beginning of therapy. An
interesting statistical method consists in integrating a priori information in the
distribution of the parameters, learned from a given database, and to combine this
information with the individual estimation from the available data on a given animal.
To do so, we integrated 10 more mice to the study to gain statistical power and then
randomly divided the population of 20 animals between two groups. One group was
used to learn the parameters distribution and the other for forecast purposes. It
should be pointed out that for a given individual, no information from that individual
was used to estimate the a priori distributions: only information from an independent
group. The full procedure was replicated 100 times to ensure statistical significance.
Predictions obtained using this technique were significantly improved when
forecasting global future growth curves based on 3 data points (Figure 5). Not all the
models equally benefited from the addition of a priori information in the fitting
procedure (Figure 5). Low-parameterized models such as power growth, exponential
𝑉0 and Gompertz, which also have the lowest parameter inter-individual variability
15
(Table 3), exhibited great benefit, while the dynamic CC model (which has three
degrees of freedom) only had modest but still significant benefit (from 19.7% to
32.4%). Indeed, due to a widely spread distribution of parameters (especially 𝐾0 , see
Table 3), the a priori distribution does not contain much information for this model,
and does not add much information to the fits. On the other hand the power growth
model, whose distribution in 𝛾 is particularly narrow (Table 3), has a much more
informative a priori distribution that translated into the most drastic improvement of
the predictive power (from 14.9% to 60.2%). The impact of the addition of the a priori
information was less important when using more data points (results not shown).
These results demonstrate that addition of a priori information in the fit procedure
greatly improves the forecast performances of the models, in particular when using
fewer data points with low-parameterized models such as the power growth model.
Discussion
Rigorous study of the descriptive and predictive power for a class of mathematical
models of tumor growth was performed. Based on a detailed quantification of the
measurement error, five of the nine models initially considered exhibited significant
descriptive power. Numerical identifiability was also considered as a criterion for
model comparison.
Derivation of a specific measure error model was a fundamental consideration in the
quantitative assessment of the models’ performances and statistical rejection of
inaccurate growth theories. As already observed by others [17–19], our results
confirmed that tumor growth cannot be continuously exponential (constant doubling
time) and consequently that it cannot be explained only by proliferation of a constant
fraction of the neoplasm, specifically at initiation. This fact has important implications
in terms of identification of the inception time of the tumor [17].
Sigmoidal models such as the Gompertz model (exponential decay of the relative
growth rate) were found to be valid descriptions of tumor growth, while logistic decay
of the relative growth rate had to be rejected. The exponential-linear model was also
rejected by our analysis, a result that contrasts with its good fitting power observed in
[5] and that can be explained by the error model we used in our analysis (volumedependent variance) that derived from our quantification of the measurement error.
Using a constant variance error model gave a much a better rank to this model.
Despite its good fitting properties, the Gompertz model does not elucidate the
underlying biology and the dynamic CC model, while being more biologically-based,
and found to have similar descriptive properties, exhibited large variability of the
parameters in the population that translated into low numerical identifiability and
ultimately limited predictive power. Nevertheless, the latter model was not designed
with the intent to quantify tumor growth, but rather to be able to describe the effects
of anti-angiogenic agents on global tumor dynamics.
16
As a result of our analysis, the power growth model (relative growth rate proportional
to a power of the volume) appears as a simple, robust, descriptive and predictive
mathematical model for murine tumor growth kinetics that has clear and simple
biological foundation. It suggests a general law of macroscopic in vivo tumor growth:
only a subset of the tumor cells proliferates, the measure of this subset being
proportional to a constant fractional power of the volume (its fractal dimension). This
model showed a close match to our data and to the Gompertz curve (in the range of
the observed volumes) and reconciles the Gompertz model with the biology by giving
a mechanistic explanation of the growth rate decay that naturally happens when
fractal dimension of the proliferative tissue is lower than 3. Of all the models we
considered, the power growth model is the most parsimonious and identifiable, as
well as the most biologically explicit, descriptive, and predictive. Moreover, it fits well
with the general concept of fractal growth that is ubiquitous in biology of growth
processes where self-similarity arises from dynamical auto-organization (examples
being trees and pulmonary or vascular development).
Nevertheless, the origin of the fractal nature of the proliferative tissue remains to be
elucidated. This should also be taken with caution when dealing with very small
volumes (at the scale of several cells for instance) for which the assumption of fractal
dimension falls, since the tumor tissue cannot be considered as a continuous
medium anymore. Additionally, our results were obtained in a particular setting
(syngeneic murine model) and, although consistent with other results involving
mammary cell line [20], remain to be confirmed and substantiated by extension to
broader experimental (in particular human) settings.
Potential use of mathematical models as forecasting tools was assessed. On top of
its very good descriptive properties and identifiability robustness, the power growth
model was found to be globally the most predictive over a wide range of situations
regarding to number of data points used and prediction depths, with an overall
median success score of 60%. In some situations, other models are more indicated,
such as the dynamic CC model when using 6 data points and predicting the next
data point (90% success).
Use of a priori information to facilitate parameter identification proved helpful,
particularly when few data were available and models had a low number of
parameters. In such a situation, it improved the success rate of the power growth
model from 14.9% to 60.2%. This comes partly from the important homogeneity of
our growth data that generated a narrow and very informative distribution of the
power growth parameter 𝛾, which in turn powerfully assisted fitting procedure. In
more practical situations such as patient data, much more heterogeneity of the
growth data can be expected.
Translating our results into a clinical setting raises the possibility of forecasting solid
tumor growth using simple macroscopic models and proposes the power growth
model in particular as a good initial candidate. Further information could and should
17
be extracted from (functional) imaging devices, feeding more complex mathematical
models that could help design more accurate in silico prediction tools [7,8].
Our analysis shows that use of mathematical models could also be a valuable tool for
helping preclinical anti-cancer research as it could lead to interesting applications for
assessing drug efficacy (for example, by comparing the treated growth curve to the
expected growth when no treatment is administered). Although integration of therapy
remains to be added (and validated) to the power growth model, more classical
models (such as exponential-linear [5] or dynamic CC [3]) have already been shown
to be able to predict cytotoxic or anti-angiogenic effects of drugs on tumor growth.
Our methods have allowed precise quantification of their respective descriptive and
predictive powers, which, in combination with the models’ intrinsic biological
foundations, could be of value when deciding among such models which best
captures the observed growth behaviors relevant preclinical settings.
Acknowledgments
We thank Etienne Baratchart for valuable suggestions and comments. This work was
supported by the National Cancer Institute under Award Number U54CA149233 (to
L. Hlatky). The content is solely the responsibility of the authors and does not
necessarily represent the official views of the National Cancer Institute or the
National Institutes of Health.
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Figure 1: Volume measurement error. A. First measured volume 𝑌1 against second
one 𝑌2 . Largest (L) and smallest (w) diameters were measured subcutaneously using
𝜋
calipers and then formula 𝑉 = 6 𝑤 2 𝐿 was used for computation of the volume
(ellipsoid). Also plotted is the regression line (correlation coefficient 𝑟 = 0.98, slope of
the regression = 0.96). B. Error 𝑌1 − 𝑌 against approximation of the volume given by
the average of the two measurement 𝑌 =
𝑌1 +𝑌2
2
. The 𝜒 2 test rejected Gaussian
distribution of constant variance (𝑝 = 0.004) C. Histogram of the normalized error
𝜎𝑌 𝛼 ,
𝜎𝑉𝑚𝛼 ,
applying the error variance model given by 𝛴 = {
𝑌 ≥ 𝑉𝑚
, with 𝛼 = 0.84 and
𝑌 < 𝑉𝑚
𝑉𝑚 = 83 𝑚𝑚3 . It shows Gaussian distribution (𝑝 = 0.196) with standard deviation 𝜎 =
0.21.
Figure 2: Descriptive power. A. Excised tumor and hematoxylin and eosin
immunostaining of the tissue resulting from in vivo growth of LLC cells. B. Illustrative
example of all growth models fitting the same individual kinetic. Errorbars correspond
to the standard deviation of the a priori estimate of measurement error. From visual
examination on this example, exponential 1, Logistic and exponential-linear are not
20
appropriate while the others describe the growth in a satisfactory manner. C.
Distributions of normalized errors (NE) for the tested models. Residuals include fits
over all the animals and all the time points. Log = generalized, Exp-L = exponentiallinear, GLog = generalized logistic, Dyn CC = dynamic CC, Gomp = Gompertz, PG =
power growth.
Figure 3: Predictive power. Illustrative example of the forecast performances of the
four best descriptive models. Six data points were used to learn the animal
parameters and predict future growth. Although all models succeeded in predicting
the next day data, exponential 𝑉0 and Gompertz failed for global forecast of the future
while power growth and dynamic CC succeeded, for this animal.
Figure 4: Prediction depth. Test of the predictive power of the models depending
on the number of data points used and the prediction depth in the future. At position
(𝑁, 𝑑) the color represents percentage of successfully predicted animals when using
𝑁 data points and forecasting the 𝑁 + 𝑑 -th data. Only data where 𝑁 + 𝑑 ≤ 10 was
considered of interest since few animals had more than 10 longitudinal
measurements. First line is model with two parameters and second line models with
three parameters.
Figure 5: A priori information. Global group of 20 animals was randomly divided
into two subgroups. One was used to retrieve the models parameters distribution in
the population. This a priori information was then combined to the individual
information from fitting 𝑁 = 3 data points in order to predict all the future growth. The
full procedure was repeated 100 times. For a given individual, prediction was
considered successful when mean normalized error was lower than 3. A. Success
rates of the models over all replicates when using a priori information on the
parameters population distributions, 𝑁 = 3 data points and predicting the global
future growth curve (mean ± standard deviation). Improvement of the prediction is
statistically significant in all situations (𝑝 < 10−18 by Student’s t-test) B. Illustrative
example of the benefit of adjunction of a priori information for a given mouse, using
the power growth model for prediction.
Supporting Information Legends
Supplementary Figure 1: Relative growth rates. Comparison of the time decrease
of growth rates the considered models. Parameters are the ones resulting from the
individual fit of tumor growth curve of an individual mouse. Note that the five best
descriptive models (Gompertz, dynamic CC, power growth and generalized logistic)
have very similar profiles while the others exhibit qualitatively different behavior.
Supplementary Figure 2: Individual predictions of future growth using 6 data
points and the dynamic CC model. Individual parameters were estimated using the 6
first data points and future growth is extrapolated. Based on the criterion of a median
21
NE smaller than 3 for the total future prediction (meaning that the median model
prediction is within 3 standard deviations of the measurement error), 7 tumor growths
were considered to be reasonably predicted, corresponding to animals 1, 2, 5, 7, 8, 9
and 10. Prediction of only the next data point was considered successful for all but
animal 4.
Supplementary Figure 3: Forecast improvement of the Power Growth model
when using a priori information. Fits are performed using the first three data
points, for each animal. A priori information (learned on a different data set) is added
during the fit procedure for the predictions on the right. Shown is a particular replicate
among the 100 subdivisions of the global group (20 mice) into one “learning” group
and one “forecast” group.
Model
Power growth
Identifiability
score (%)
100
Gompertz
99.8
Dynamic CC
61.5
Generalized
logistic
1.19
Exponential 𝑉0
100
Exponential-linear
95.3
Logistic
Exponential 1
100
100
Par.
Score (%)
𝑎
𝐾
𝑎
𝑏
𝐾0
𝑎
𝐾
𝛼
𝑎0
𝑎1
-
100
99.8
86.4
94.0
75.1
43.1
67.6
25.5
95.3
100
-
Median
Dev. (%)
0.001
0.96
0.4
2.28
87.2
72.5
390
0.0013
-
Table 1: Robustness of the models’ numerical identifiability. Numerical identifiability of
the models was assessed by systematically varying the initial condition of the optimization
algorithm in a range of diameter 4 standard deviations in each parameter direction. We then
tested agreement with the parameter sets obtained with base value of the initial guess. An
identifiability score was then derived by computing the proportion of fits giving different
convergence of the minimization, the difference being defined by a relative deviation larger
than 10%. The identifiability score reported is the proportion of success, among the 𝑁 𝑝
individual fits where 𝑝 is the number of parameters and 𝑁 is the number of meshes in each
parameter direction (here 𝑁 = 11). When lower than 100%, further analysis was performed
and the same score was computed for each parameter of the model. We also report their
median relative deviation. Par. = Parameter. Dev. = Deviation
22
p
AIC
mNE
mR2
mRMSE
J
Number of
parameters
Dynamic CC
0.721
1054
0.81
0.98
1.73
9.38
3
Gompertz
0.636
1069
0.74
0.98
1.83
12.6
2
Power growth
0.849
1070
0.87
0.98
1.72
10.8
2
Generalized
logistic
0.427
1071
0.89
0.98
2.15
14.6
3
Exponential
𝑉0
<0.001
1081
1.23
0.95
2.02
14.7
2
Logistic
<0.001
1098
1.63
0.96
2.90
28.9
2
Exponentiallinear
<0.001
1154
1.46
0.96
2.80
27.6
2
Exponential 1
<0.001
1267
5.78
0.63
6.36
147
1
Model
Table 2: Fitting performances of growth models. p = p-value of the 𝜒 2 test for normal
distribution of residuals with standard deviation 𝜎 = 0.21. AIC = Akaike Information Criterion
computed using Monolix (proportional power variance error model). mNE = median
normalized error (over all time points and animals). Normalized error is defined by 𝑁𝐸𝑖 =
𝑌𝑖 −𝑓(𝑡𝑖 ,𝛽)
|
𝛴𝑖
| for model 𝑓, parameter set 𝛽, time point 𝑡𝑖 , data 𝑌𝑖 and 𝛴𝑖 defined by formula (2).
mR2 = coefficient of determination averaged over all the individual fits. mRMSE = Root Mean
Squared Errors averaged over all the individual fits. J = total sum of squared errors upon all
the individuals.
23
Model
−𝟏
Power growth
Gompertz
Dynamic CC
Generalized
logistic
Exponential 𝑉0
𝜸−𝟏
Parameters (CV)
)
𝜸
𝒂 (𝒅𝒂𝒚 𝒎𝒎
1.11 (29)
0.74 (6.5)
𝒂 (𝒅𝒂𝒚−𝟏 )
𝑲 (𝒎𝒎𝟑 )
8.87 × 10−2 (26)
8.92 × 103 (67)
𝟑
𝒂 (𝒅𝒂𝒚−𝟏 )
𝑲𝟎 (𝒎𝒎𝟑 )
𝒃 (𝒎𝒎𝟐 ⋅ 𝒅𝒂𝒚−𝟏 )
9.33 × 10−1 (109)
2.91 (43)
44.8 (115)
𝜶
𝒂 (𝒅𝒂𝒚−𝟏 )
𝑲 (𝒎𝒎𝟑 )
3
158 (78)
7.49 × 10 (38)
0.055 (237)
−𝟏
𝒂 (𝒅𝒂𝒚 )
𝑽𝟎 (𝒎𝒎𝟑 )
0.23 (7)
17.0 (29)
Table 3: Parameter values for the four best descriptive models + exponential 𝑽𝟎 .
Shown is the mean value within the population and in parenthesis the coefficient of variation
(CV, defined as the (standard deviation divided by mean and multiplied by 100) that
quantifies inter-individual variability.
Model
Power growth
Dynamic CC
Exponential 𝑉0
Gompertz
Overall
median
success (%)
60
55
45
27.5
Overall
mean
success
(%)
48.3
51.3
42.8
36.8
𝑺𝟔𝒈𝒍𝒐𝒃 (%)
𝒎𝑵𝑬𝟔𝒈𝒍𝒐𝒃
𝑺𝟔𝟏 (%)
60
70
30
40
2.22
1.89
4.24
3.92
70
90
60
80
Table 4: Forecast properties of four growth models having equivalent descriptive
power plus exponential growth. Prediction was considered successful when normalized
error NE (or median NE when prediction involves more than one data point) was lower than
3.The first column is global quantification of the predictive power over all the number of data
points used (from 3 to 7) and all the prediction depths (up to 7). The last three columns are
quantifications of the predictive goodness for one given number of data points (N=6).
6
Predictions were assessed on the remaining data points. 𝑆𝑔𝑙𝑜𝑏
= success rate for prediction
6
of the global remaining curve, based on 6 data points (total of 10 mice). 𝑚𝑁𝐸𝑔𝑙𝑜𝑏
= median
normalized prediction error among all the future growths of all individuals. 𝑆16 = success rate
for prediction only of the next data point, i.e. a prediction depth of one.
24
Model
Power growth
Gompertz
Dynamic CC
Generalized
logistic
Parameter
𝑎
𝛾
𝑎
𝐾
𝑎
𝑏
𝐾0
𝑎
𝐾
𝛼
Initialization
1
2/3
0.1
10000
3
0.5
10
10
10000
0.01
0.1
Exponential 𝑉0
𝑎
𝑉0
Exponential-linear
𝑎0
𝑎1
Logistic
𝑎
𝐾
10000
Exponential 1
𝑎
0.1
20
0.2
500
1
Supplementary Table 1: Initializations of the least squares minimization algorithm
25